\section{Results}\label{sec:results}
For the training session of the models, we resort to a standard $5$-Fold Cross-Validation process, performing hyper-parameter tuning through a grid search mechanism. This procedure is particularly beneficial to infer the correct number of clusters that best approximates the data distribution. Indeed, the sole maximization of certain metrics would often result in the trivial solution of adopting as many clusters as possible, thus overfitting data.\\
We guarantee the consistency of each tested combination by performing five (in the case of probabilistic models) to ten re-initializations of the initial configuration, given the relevant importance assumed by the starting phase.\\
We collect a set of different metrics to evaluate the quality of the clustering. Unfortunately, since we act in a completely unsupervised context, where adopting any sort of ground truth would be biased by the underlying modeling assumption. For this reason, the Météo-France datasets is adopted as a monitor of the performances of the clustering algorithms, however without being involved in the model selection process. Hence, we can only recur to the so-called \textit{internal} clustering scores (in contrast to \textit{external} metrics, leveraging some knowledge on some labelled data). These measures embed concepts of intra and inter-cluster dispersions, in order to evaluate the goodness of the assignment of each sample.
\begin{itemize}
    \item \textit{Inertia} or \textit{Sum of Squared Errors (SSE)}: it is the typical measure when evaluating the clustering configurations. The measure indicates a sum of all the squared residuals between each point, and the cluster's centroid to which the point is assigned:
    $$ I = \sum_{k=1}^K \sum_{n=1: \ z_{nk} = 1}^N (x_n - \bar{x}_k)^2$$
    \item \textit{Silhouette Score}: a measure ranging in the interval $[-1, 1]$, evaluating the position of each sample with respect to the cluster which is assigned to, and the closest cluster. For each point it is evaluated:
    \begin{itemize}
        \item \textit{Mean dissimilarity} between a point $x_i$ and the nearest cluster $C_k$:
        $$ b(x_i) = \min_{k\neq i} \sum_{j \in C_k} d(i,j)$$
        \item \textit{Mean intra-cluster distance} between a point $x_i$ and all the other points belonging to its cluster:
        $$ a(x_i) = \frac 1 {|C_i| -1} \sum_{j \neq i} d(i,j)$$
    \end{itemize}
    The silhouette score for a point $x_i$ is then:
    $$ s(x_i) = 
    \begin{cases}
    0 & \text{if }|C_i| = 1\\
    \frac{b(x_i) - a(x_i)}{\max\{a(x_i), b(x_i)\}} & \text{otherwise}
    \end{cases}$$
    The overall silhouette score is then:
    $$ \mathrm{Silhouette} = \max_k \bar{s}_k$$
    i.e., the maximum between the mean scores per cluster
    \item \textit{Calinski-Harabasz (CH) Index}: a measure keeping into account the intra and inter-clusters variations
    $$ CH(k) = \frac{B(k)}{W(k)}\cdot \frac{n-k}{k-1}$$
    where:
    \begin{itemize}
        \item $B(k)$ is the between-cluster variation, having $k-1$ degrees of freedom
        \item $W(k)$ is the within-cluster variation, having $n-k$ degrees of freedom
    \end{itemize}
    \item \textit{Bayesian Information Criterion (BIC)} \parencite{Falkena2020RevisitingTI}: defined under the maximum likelihood estimation framework, it is evaluated as:
    $$ BIC = k \log N - 2\log \hat{L}, \quad \hat{L} = p(x| \hat{\theta}_{MLE})$$
    \item \textit{Evidence Lower Bound (ELBO)}: equivalent to the pre-defined variational objective
\end{itemize}
Table \ref{tab:metrics} summarizes the different metrics.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|} 
 \hline
 \cellcolor[rgb]{1, 0.87, 0.4}
 \textbf{Model} & \thead{\textbf{Inertia}\\$\downarrow$} & \thead{\textbf{Silhouette}\\\textbf{score}\\$\uparrow$} & \thead{\textbf{Calinski}\\\textbf{Harabasz}\\\textbf{Index}\\$\uparrow$} & 
 \thead{\textbf{Bayesian}\\\textbf{Information}\\\textbf{Criterion}\\$\uparrow$} &
 \thead{\textbf{ELBO}\\$\uparrow$}\\
 \hline
 \cellcolor[rgb]{1, 0.96, 0.8} \textbf{K-Means} & \checkmark & \checkmark & \checkmark & \checkmark & \\ 
 \cellcolor[rgb]{1, 0.96, 0.8} \textbf{GMM} & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark\\
 \cellcolor[rgb]{1, 0.96, 0.8} \textbf{DPMM} & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark\\
 \hline
\end{tabular}
\caption{Evaluation Metrics for each model. $\uparrow$/$\downarrow$ indicates that the score has to be maximized/minimized}
\label{tab:metrics}
\end{table}
\subsection{Theoretical Results}
The process of Cross-Validation allows to select the best performing models. Table \ref{tab:best_models} reports the results for each algorithm, under each dimensionality reduction technique, including which score resulted the optimized objective.\\
At this point, we encounter the first critical points of this modeling step:
\begin{itemize}
    \item The two dimensionality reduction techniques cannot be compared just looking at the absolute values of the score. Being dependent by concept of dispersion in the space, it is clear that a reduced space mapped on a less sparse region would result in higher scores. Due to a minor dimensionality of the feature space, the VAE-reduced space is more dense than the PCA hyper-space, thus leading to lower absolute scores
    \item The scores are a good metric to select the best model across several configurations of the same algorithm. However, as proven consequently, they fail in objectively determining which model, among the different assumptions, should be adopted as the reference candidate. While the two Mixture models are a closer assumption, the same does not happen when comparing K-Means with the probabilistic frameworks. With the exception of the ELBO, these metrics require a ``deterministic'' setting, like an hard thresholding of the assignments, thereby partially mitigating the generalization power of the probabilistic models
\end{itemize}
\begingroup
\setlength{\LTleft}{-20cm plus -1fill}
\setlength{\LTright}{\LTleft}
\begin{longtable}[!htbp]{c|
>{\columncolor[HTML]{FFF5CC}}c |c|c|c|c|c|}
\cline{2-7}
\multicolumn{1}{l|}{} &
  \cellcolor[HTML]{FFDE66}\textbf{Model} &
  \textbf{Objective} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Inertia\\ $\downarrow$\end{tabular}} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Silhouette\\ Score\\ $\uparrow$\end{tabular}} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Calinski\\ Harabasz\\ Index\\ $\uparrow$\end{tabular}} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Bayesian\\ Information\\ Criterion\\ $\uparrow$\end{tabular}} \\ \hline
\endfirsthead
%
\multicolumn{7}{c}%
{{\bfseries Table \thetable\ continued from previous page}} \\
\cline{2-7}
\multicolumn{1}{l|}{} &
  \cellcolor[HTML]{FFDE66}\textbf{Model} &
  \textbf{Objective} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Inertia\\ $\downarrow$\end{tabular}} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Silhouette\\ Score\\ $\uparrow$\end{tabular}} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Calinski\\ Harabasz\\ Index\\ $\uparrow$\end{tabular}} &
  \textbf{\begin{tabular}[c]{@{}c@{}}Bayesian\\ Information\\ Criterion\\ $\uparrow$\end{tabular}} \\ \hline
\endhead
%
\hline
\endfoot
%
\endlastfoot
%
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}} &
  \textbf{K-Means} &
  CH &
  $5.27 \times 10^{11}$ &
  $0.113$ &
  $93.4$ &
  $-1.2\times 10^{-5}$ \\
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}} &
  \textbf{GMM} &
  Silhouette &
  $5.71 \times 10^{11}$ &
  $0.076$ &
  $67.4$ &
  $-1.2\times 10^{-5}$ \\
\multicolumn{1}{|c|}{\multirow{-3}{*}{\cellcolor[HTML]{FFDE66}\textbf{PCA}}} &
  \textbf{DPMM} &
  BIC &
  $5.77 \times 10^{11}$ &
  $0.060$ &
  $62.8$ &
  $-1.2\times 10^{-5}$ \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}} &
  \textbf{K-Means} &
  CH &
  $4.66 \times 10^7$ &
  $0.268$ &
  $233.6$ &
  $-2.46\times10^{-4}$ \\
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}} &
  \textbf{GMM} &
  CH &
  $5.24 \times 10^7$ &
  $0.199$ &
  $180.8$ &
  $-2.49\times10^{-4}$ \\
\multicolumn{1}{|c|}{\multirow{-3}{*}{\cellcolor[HTML]{FFDE66}\textbf{VAE}}} &
  \textbf{DPMM} &
  Silhouette &
  $5.25 \times 10^7$ &
  $0.199$ &
  $180$ &
  $-2.49\times10^{-4}$ \\ \hline
\caption{Best performing models, under the two dimensionality reductions. $\uparrow$/$\downarrow$ indicates that the score has to be maximized/minimized. The column ``Objective" indicates which score was optimized to in the Cross-Validation process of the model}
\label{tab:best_models}\\
\end{longtable}
\endgroup
In Table \ref{tab:best_models} are reported only 4-clusters models, as we believe they are the correct trade-off to describe the regimes' dynamics, while not losing in generalization power. However, some 7-clusters configurations (Figure \ref{fig:kmeans7}) under the K-Means choice are obtained from the model selection experiments, supporting the results of some previous studies \parencite{grams}. Figures \ref{fig:kmeans4} report the centroids configurations obtained with K-Means with the PCA (Figure \ref{fig:pca_kmeans4}) and VAE (\ref{fig:vae_kmeans4}) reduction schemes.
\begin{figure}[!htbp]
    \centering
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/pca_kmeans4.png}
        \caption{Visualization of the centroids after clustering with K-Means on PCA-reduced dataset}
        \label{fig:pca_kmeans4}     
    \end{subfigure}
    \hfill
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/vae_kmeans4.png}
        \caption{Visualization of the centroids after clustering with K-Means on VAE-reduced dataset}
        \label{fig:vae_kmeans4}
    \end{subfigure}
    \caption{K-Means with 4 clusters. AR (top left), NAO+ (top right), NAO- (bottom left), SB (bottom right)}
    \label{fig:kmeans4}
\end{figure}
The inferred clusters' centers are very close to the theoretical ones, on the one hand confirming how globally K-Means could be adequate in describing the average dynamics. However, as Table \ref{tab:freq_kmeans} reports, a disagreement exists between the computed frequencies of the regimes in literature \parencite{2011ClDy...36...19C,van_der_Wiel_2019, WeatherRegimeTransitionsandtheInterannualVariabilityoftheNorthAtlanticOscillationPartIALikelyConnection, meteofrance} and our experiments on K-Means. In particular, the major differences are highlighted into the configuration modeling on the VAE mapping.
\begin{table}[!htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{c|c|c|c|c|c|}
\cline{2-6}
                                                           & \cellcolor[HTML]{FFDE66}\textbf{Model}   & \textbf{AR} & \textbf{NAO+} & \textbf{NAO-} & \textbf{SB} \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{PCA}} & \cellcolor[HTML]{FFF5CC}\textbf{K-Means} & $20.6$\%   & $33.2$\%     & $19.9$\%     & $26.3$\%   \\ \cline{1-1}
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{VAE}} & \cellcolor[HTML]{FFF5CC}\textbf{K-Means} & $17.5$\%   & $47.4$\%     & $13.9$\%     & $21.2$\%   \\ \hline
\multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{2011ClDy...36...19C}}}         & $23.3$\%   & $29.9$\%     & $22.4$\%     & $24.5$\%   \\
\multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{van_der_Wiel_2019}}}        & $20$\%   & $33$\%     & $20$\%     & $28$\%   \\
\multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{WeatherRegimeTransitionsandtheInterannualVariabilityoftheNorthAtlanticOscillationPartIALikelyConnection}}} & $21.8$\% & $31.9$\% & $18.1$\% & $28.2$\% \\
\multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{meteofrance}}} & $20.4$\% & $31.8$\% & $18.4$\% & $29.4$\% \\ \hline
\end{tabular}%
}
\caption{Frequencies of the regimes as predicted by \parencite{2011ClDy...36...19C,van_der_Wiel_2019, WeatherRegimeTransitionsandtheInterannualVariabilityoftheNorthAtlanticOscillationPartIALikelyConnection, meteofrance} vs. Frequencies as predicted by K-Means under PCA and VAE reduction schemes. We notice how K-Means is highly sensitive to the inferred mapping of VAE}
\label{tab:freq_kmeans}
\end{table}

\begin{figure}[!htbp]
    \centering
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/pca_kmeans7.png}
        \caption{Visualization of the centroids after clustering with K-Means on PCA-reduced dataset. The fine-tuning in this case emitted 7 clusters as the best configuration. Other than the four main standard configurations, it is possible to identify: SB- (top right), AR - (second row, on the right). The AR configuration can instead be conceived as average case between the top-left and third row, on the left plots}
        \label{fig:pca_kmeans7}
    \end{subfigure}
    \hfill
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/vae_kmeans7.png}
        \caption{Visualization of the centroids after clustering with K-Means on VAE-reduced dataset. The fine-tuning in this case emitted 7 clusters as the best configuration. Other than the four main standard configurations, it is possible to identify: SB- (top right), AR - (second row, on the right). The AR configuration can instead be conceived as average case between the top-left and third row, on the left plots}
        \label{fig:vae_kmeans7}
    \end{subfigure}
    \caption{K-Means with 7 clusters}
    \label{fig:kmeans7}
\end{figure}
Focusing instead on the predicted centroids of the Mixture Models (Figure \ref{fig:mm4}) we immediately recognize some differences, with respect to K-Means, in the minor configurations, namely: AR and SB. This is a critical point we encounter, which is still more highlighted in the next section, when compared to Météo-France predictions, suggesting a small bias of these modeling choices.

\begin{figure}[!htbp]
    \centering
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/pca_gmm4.png}
        \caption{Visualization of the centroids after clustering with GMM on PCA-reduced dataset}
        %However, we notice some minor differences in the anomaly magnitudes, especially referring to the positive North Atlantic Oscillation. This is probably due to the complexity of adapting 15-dimensional distributions to data.}
        \label{fig:pca_gmm4}
    \end{subfigure}
    \hfill
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/vae_gmm4.png}
        \caption{Visualization of the centroids after clustering with GMM on VAE-reduced dataset}
        %Compared to the PCA-reduction, the regimes are more in line with the theoretical expectations, especially looking at the magnitudes of the anomaly pressure levels.}
        \label{fig:vae_gmm4}
    \end{subfigure}
    \vfill
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/pca_bayesgmm4.png}
        \caption{Visualization of the centroids after clustering with DPMM on PCA-reduced dataset}
        %Again, we report an overall good behavior, except that results tend to be noisier. As for PCA-based GMM, the DP-Mixtures probably likely strive to adapt high-dimensional data.}
        \label{fig:pca_bayesgmm4}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/vae_bayesgmm4.png}
        \caption{Visualization of the centroids after clustering with DPMM on VAE-reduced dataset}
        %As for VAE-based GMM, the Dirichlet Process Mixtures benefit from the low-dimensional mapping and they are able to correctly identify the configurations.}
        \label{fig:vae_bayesgmm4}
    \end{subfigure}
    \caption{Mixture Models with 4 clusters. AR (top left), NAO+ (top right), NAO- (bottom left), SB (bottom right)}
    \label{fig:mm4}
\end{figure}

Again, we recur to an a-posteriori validation with the expected frequencies, as reported by the literature. Table \ref{tab:freq_mm} confirms that Mixture Models tend to benefit more from the robust reduction obtained through the $\sigma$-VAE, also driven by the major ease of fitting multi-variate distributions on a minor dimensionality of the space.
\begin{table}[!htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{c|
>{\columncolor[HTML]{FFF5CC}}c |c|c|c|c|}
\cline{2-6}
 &
  \cellcolor[HTML]{FFDE66}\textbf{Model} &
  \textbf{AR} &
  \textbf{NAO+} &
  \textbf{NAO-} &
  \textbf{SB} \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}} &
  \textbf{GMM} &
  $27.9$\% &
  $38.5$\% &
  $15.5$\% &
  $18.1$\% \\
\multicolumn{1}{|c|}{\multirow{-2}{*}{\cellcolor[HTML]{FFDE66}\textbf{PCA}}} &
  \textbf{DPMM} &
  $27.1$\% &
  $39.9$\% &
  $14.8$\% &
  $18.3$\% \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}} &
  \textbf{GMM} &
  $17.7$\% &
  $31.6$\% &
  $26.9$\% &
  $23.8$\% \\
\multicolumn{1}{|c|}{\multirow{-2}{*}{\cellcolor[HTML]{FFDE66}\textbf{VAE}}} &
  \textbf{DPMM} &
  $24.9$\% &
  $32.3$\% &
  $18.1$\% &
  $24.8$\% \\ \hline
\multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{2011ClDy...36...19C}}} & $23.3$\% & $29.9$\% & $22.4$\% & $24.5$\% \\
\multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{van_der_Wiel_2019}}} &
  $20$\% &
  $33$\% &
  $20$\% &
  $28$\% \\
  \multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{WeatherRegimeTransitionsandtheInterannualVariabilityoftheNorthAtlanticOscillationPartIALikelyConnection}}} & $21.8$\% & $31.9$\% & $18.1$\% & $28.2$\% \\
\multicolumn{2}{|c|}{\cellcolor[HTML]{EFEFEF}\textbf{\parencite{meteofrance}}} & $20.4$\% & $31.8$\% & $18.4$\% & $29.4$\% \\ \hline
\end{tabular}%
}
\caption{Frequencies of the regimes as predicted by \parencite{2011ClDy...36...19C,van_der_Wiel_2019, WeatherRegimeTransitionsandtheInterannualVariabilityoftheNorthAtlanticOscillationPartIALikelyConnection, meteofrance} vs. Frequencies as predicted by Mixture Models under PCA and VAE reduction schemes. We notice how under PCA statistics tend to be more misaligned}
\label{tab:freq_mm}
\end{table}

The soft-assignments allow us to infer the intermediate transitions, thus excluding the necessity of adopting models with higher numbers of clusters \parencite{2011ClDy...36...19C, grams}. Table \ref{tab:7regimes} reports the frequency where two regimes are mixed together with similar probabilities (i.e. when they account for at least 70\% of the total assigned probabilities). The results are quite realistic, since, empirically, positive NAO and SB are likely to merge as it will be also more deeply outlined in Sections \ref{sec:transitions} and \ref{sec:physical}.

\begin{table}[!htbp]
\centering
\resizebox{.6\textwidth}{!}{%
\begin{tabular}{
>{\columncolor[HTML]{FFDE66}}c |c|c|c|c|}
\cline{2-5}
\cellcolor[HTML]{FFFFFF}\textbf{}                           & \cellcolor[HTML]{FFDE66}\textbf{AR} & \cellcolor[HTML]{FFDE66}\textbf{NAO+} & \cellcolor[HTML]{FFDE66}\textbf{NAO-} & \cellcolor[HTML]{FFDE66}\textbf{SB} \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{AR}}   & -                                   & 2.4\%                                 & 2.4\%                                 & 2.6\%                               \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}} & 2.4\%                               & -                                     & 3\%                                   & \textbf{6.3\%}                               \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}} & 2.4\%                               & 3\%                                   & -                                     & 3.3\%                               \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{SB}}   & 2.6\%                               & \textbf{6.3\%}                                 & 3.3\%                                 & -                                   \\ \hline
\end{tabular}%
}
\caption{Frequencies (\%) of the intermediate transitions between pairs of regimes, as predicted by our Mixture Models.}
\label{tab:7regimes}
\end{table}

Finally, as for K-Means, some configurations of Dirichlet Process Mixture Model allocate mass on 7-clusters scenarios (Figure \ref{fig:bayesgmm7}), resulting a sub-optimal solution under the illustrated scores. Differently than K-Means though, the 7 regimes results to be only partially descriptive of the 7 theorized regimes of \parencite{grams}, clearly representing noisy intermediate transitions. While this experiment is not further studied, we believe it could be of future interest analyzing these 7 clusters, possibly explaining at a finer-grained level some perspectives on weather regimes.
\begin{figure}[!htbp]
    \centering
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/pca_bayesgmm7.png}
    \end{subfigure}
    \hfill
    \begin{subfigure}{0.49\textwidth}
        \centering
        \includegraphics[width = .95\textwidth]{imgs/vae_bayesgmm7.png}
    \end{subfigure}
    \caption{Variational Dirichlet Process Mixture Model with 7 clusters}
    \label{fig:bayesgmm7}
\end{figure}
\subsection{Comparison with Météo-France}
During our empirical validation of random sampled predictions we notice some discrepancies referred to K-Means predictions. However such analysis would not be feasible when enlarged to the entire dataset.
Hence, we resort to a thorough comparison with Météo-France historical predictions, allowing us to circumscribe our modeling choices for the successive tasks. The models we select are K-Means under PCA reduction, and the Mixture Models under the VAE pre-processing.  As Figure \ref{fig:conf_matrix} reports, K-Means shows consistent performances, since it is aligned with the same modeling assumption of Météo-France.
\begin{figure}[!htbp]
    \centering
    \includegraphics[width = .85\textwidth]{imgs/conf_matrix.png}
    \caption{Confusion matrices between our 4 cluster models and Metéo-France historical predictions. The Mixture Models are inline with the agency's predictions. K-Means algorithm produces better scores, since it is aligned with the modeling choice of Meteo-France. From this illustration, it is evidenced how each model mostly strives with the Atlantic Ridge (AR) and Scandinavian Blocking (SB) configurations}
    \label{fig:conf_matrix}
\end{figure}
Mixture Models tend to be inline with the provider's data, however evidencing some insights:
\begin{itemize}
    \item North Atlantic Oscillations (NAOs) configurations present an higher True Positive Rate (TPR). This is coherent if we think that they typically correspond to the major source of variability (i.e., both EOF-1 in Figure \ref{fig:eofs}, and the first principal component would be enough to distinguish the two regimes) and they are the most geographically widespread clusters
    \item Atlantic Ridge (AR) confirms to be a ``class'' which is difficult to be completely decoupled from the other regimes. Again, the geographical configuration, and the visual results provided in Figure \ref{fig:tsne} support this idea 
\end{itemize}
\begin{figure}[!htbp]
    \centering
    \includegraphics[width = \textwidth]{imgs/roc_curves.png}
    \caption{ROC curves of the three models: K-Means  with PCA reduction (left), DPMM (center) and GMM (right) with VAE reduction}
    \label{fig:roc_curves}
\end{figure}
Further, we build the Receiver Operating Characteristic (ROC) curves of the three models. Figure \ref{fig:roc_curves} highlights the reduced performances of K-Means, compared to the probabilistic counterpart.\\
Since the Gaussian mixtures reach slighlty higher performances, the results obtained in the subsequent sections will be based on this modeling assumption.\\
Finally, zooming out to an historical overview of the incidence of regimes by winter (Figure \ref{fig:counts_winter}), Météo-France and GMM share some similarities across the years, suggesting that our framework is coherent.\\
Again, we want to emphasize how our procedure tends to be superior under a computational efficiency point of view. We overcome the constraint of using ensemble models to track the chaotic nature of the regimes, simply focusing on a more complex and calibrated mathematical assumption.

\begin{figure}
    \centering
    \begin{subfigure}{\textwidth}
        \centering
        \includegraphics[width = \textwidth]{imgs/count_winter_MF.png}
        \caption{Metéo-France}
        \label{fig:count_winter_MF}
    \end{subfigure}
    \begin{subfigure}{\textwidth}
        \centering
        \includegraphics[width = \textwidth]{imgs/count_winter_GMM.png}
        \caption{GMM}
        \label{fig:count_winter_GMM}
    \end{subfigure}
    \caption{Historical counts of winter-days by regime}
    \label{fig:counts_winter}
\end{figure}

\subsection{From Statistical To Physical Properties}\label{sec:physical}
The advantage of adopting probabilistic models is that clusters represent tractable distributions (Figure \ref{fig:distributions}), which can be analyzed by means of analytically available statistics. We compute the Kullback-Leibler (KL) divergence between centroid distributions of the different clusters, from which we are able to derive some interesting physical considerations.

\begin{figure}[!htbp]
    \centering
    \includegraphics[width =.9\textwidth]{imgs/mm_dists.png}
    \caption{Visualization of the distributions of the clusters of the two mixture models along the two main modes of variability}
    \label{fig:distributions}
\end{figure}

Roughly speaking, the KL-divergence $\mathbb{KL}(p\|q)$ of a distribution $p$ from a distribution $q$ can be interpreted as the distance of an empirical distribution, obtained from independent samples drawn from $q$, to the distribution $p$. In other words, it maps how much samples from the target distribution $q$ can be identified under the different probabilistic space of distribution $p$. It is then easy to understand that the opposite undergoes to a different relationship, and thus why the KL-divergence is not symmetric (i.e. $\mathbb{KL}(p\|q) \neq \mathbb{KL}(q\|p)$).
\begin{table}[!htbp]
\centering
\makebox[0pt][c]{\parbox{1.1\textwidth}{%
\begin{minipage}{0.48\hsize}
\centering
\begin{tabular}{
>{\columncolor[HTML]{FFDE66}}c |c|c|c|c|}
\cline{2-5}
\multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}\textbf{$\mathbb{KL}(p \| q)$}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{AR}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{SB}} \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{AR}}                   & 0                                                        & \textbf{16}                                                & 5.7                                                        & 5.9                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}}                 & 3.7                                                      & 0                                                          & 3.9                                                        & 2.5                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}}                 & 5.7                                                      & \textbf{16}                                                & 0                                                          & \textbf{16}                                              \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{SB}}                   & 5.3                                                      & 10                                                         & 2.5                                                        & 0                                                        \\ \hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{0.48\hsize}
\centering
\begin{tabular}{
>{\columncolor[HTML]{FFDE66}}c |c|c|c|c|}
\cline{2-5}
\multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}\textbf{$\mathbb{KL}(p \| q)$}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{AR}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{SB}} \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{AR}}                                  & 0                                                        & \textbf{15}                                                & 5                                                          & 8.5                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}}                                & 3.7                                                      & 0                                                          & 4.1                                                        & 2.5                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}}                                & 6.2                                                      & \textbf{15}                                                & 0                                                          & \textbf{21}                                              \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{SB}}                                  & 4.9                                                      & 9.8                                                        & 2.5                                                        & 0                                                        \\ \hline
\end{tabular}
\end{minipage}
}}
\medskip
\caption{KL-divergences of the regimes' distributions in GMM (left) and DPMM (right)}
\label{tab:KL}
\end{table}

In the context of weather regimes, we firstly notice how the two Mixture models are quite in agreement, with minor differences only related to the magnitude of the divergences. Further, we identify that AR is difficult to be completely isolated, since its configurations result quite identifiable under all the other clusters. Indeed, the column of the AR regime presents considerable low scores under the other regimes' distributions spaces. On the contrary, NAO+ patterns are well-separated from the other distributions since high scores are found for AR and NAO- configurations especially. This result is in line with physical considerations since SB, NAO- and AR are all known as ``blocked'' configurations (Scandinavian Blocking, Greenland Blocking and Atlantic Blocking), while NAO+ is a cyclonically dominated regime favouring the usual westerly flow over the European/Atlantic domain. 
We also want to highlight that $\mathbb{KL}(\textrm{NAO+}\| \textrm{SB}) = 2.5$ in both GMM and DPMM models, meaning that it is difficult to separate SB from NAO+. This situation reflects realistic physical scenarios. Indeed, several days with high probabilities for both SB and NAO+ are found.

\subsection{Transitions}\label{sec:transitions}
Ultimately, we are able to reconstruct an overview of the overall dynamics. We collect the historical transitions of the last 40 winters, as predicted by our models. To make the scenario more realistic, we consider a transition as valid only when a regime persists for at least 3 days, eventually disregarding single-day shifts interrupting the persistance of a regime. Again, the two models provide transition matrices in agreement, highlighting the aforementioned biases toward AR and NAO+ configurations. As expected, the tendency of these configurations is to persist in their state, thus justifying the higher values along the diagonal, and higher probabilities for larger-scale patterns like North Atlantic Oscillations, in accordance to \parencite{yearround}.\\
Empirically, the most usual transitions refer to the passages from NAO+ to SB, and from SB to NAO- (in agreement with \parencite{Vautard1988}), as well as between AR and NAO+ in both the two directions. At the same time, the least frequent passages include the evolution from NAO+ to NAO- \parencite{Ferranti2018}, and from NAO- to AR: all these aspects are clearly evident in our models.

\begin{table}[!htbp]
\centering
\makebox[0pt][c]{\parbox{1.2\textwidth}{%
\begin{minipage}{.5\hsize}
\centering
\begin{tabular}{
>{\columncolor[HTML]{FFDE66}}c |cccc|}
\cline{2-5}
\multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}\backslashbox{\textbf{From}}{\textbf{To}}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{AR}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{SB}} \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{AR}}                    & 77\%                                                     & \textbf{9}\%                                                        & 8\%                                                        & 7\%                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}}                  & 4\%                                                      & 88\%                                                       & 3\%                                                        & \textbf{5}\%                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}}                  & 3\%                                                      & \textbf{6}\%                                                        & 86\%                                                       & 5\%                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{SB}}                    & 5\%                                                      & \textbf{7}\%                                                        & \textbf{7}\%                                                        & 80\%                                                     \\ \hline
\end{tabular}
\end{minipage}%
\hfill
\begin{minipage}{.48\hsize}
\centering
\begin{tabular}{
>{\columncolor[HTML]{FFDE66}}c |cccc|}
\cline{2-5}
\multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}\backslashbox{\textbf{From}}{\textbf{To}}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{AR}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}} & \multicolumn{1}{l|}{\cellcolor[HTML]{FFDE66}\textbf{SB}} \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{AR}}                    & 83\%                                                     & \textbf{6}\%                                                        & 5\%                                                        & \textbf{6}\%                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO+}}                  & 4\%                                                      & 89\%                                                       & 2\%                                                        & \textbf{6}\%                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{NAO-}}                  & 4\%                                                      & \textbf{7}\%                                                        & 84\%                                                       & 5\%                                                      \\ \hline
\multicolumn{1}{|c|}{\cellcolor[HTML]{FFDE66}\textbf{SB}}                    & 7\%                                                      & \textbf{8}\%                                                        & 4\%                                                        & 81\%                                                     \\ \hline
\end{tabular}
\end{minipage}
}}
\medskip
\caption{Transition Probabilities as predicted by GMM (left) and DPMM (right)}
\label{tab:transitions}
\end{table}


%It appears that the two most persistent regimes are NAO+ and NAO-. It is in agreement with what it has been found in \cite{yearround}
%The 4 main transitions found for the two models are:
%- from AR to NAO+, 
%- from NAO+ to SB,
%- from NAO- to NAO+,
%- from SB to NAO+
%Both models agree that the less likely transitions are: from NAO+ to NAO- and from NAO- to AR.  in Cassou ?
%The preferred transitions between regimes have been shown to be NAO+ to SB to NAO- \cite{MultipleWeatherRegimesovertheNorthAtlanticAnalysisofPrecursorsandSuccessors} 